Sequences, Explicit & Recursive — Practice
A sequence is a function whose inputs are the counting numbers. Describe it from the start, or from the term before.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Arithmetic sequence — A list that grows by adding the same amount each time.
- Domain — The set of allowed inputs.
- Sequence — An ordered list of numbers. The nth one is an.
- Explicit formula — A rule giving an directly from n.
- Recursive formula — A rule giving an from an-1, plus a starting value.
- Continuous — A function you could draw without lifting the pen — no holes, jumps or blow-ups.
Problem 1
Classify 3, 7, 11, 15, … as arithmetic or geometric.
Worked answer
arithmetic
Difference is 4 each time
So arithmetic
Problem 2
Classify 2, 10, 50, 250, … as arithmetic or geometric.
Worked answer
geometric
Ratio is 5 each time
So geometric
Problem 3
Find the 12th term of 4, 7, 10, ….
Worked answer
37
4 + 11(3)
= 4 + 33 = 37
Problem 4
Find the 5th term of 3, 6, 12, ….
Worked answer
48
3 · 24
= 3 × 16 = 48
Problem 5
Write an explicit formula for 6, 10, 14, 18, ….
Worked answer
a_n = 6 + 4(n - 1)
an = 6 + (n-1)4
Or 4n + 2
Problem 6
Write a recursive formula for 5, 8, 11, 14, ….
Worked answer
a_1 = 5, a_n = a_(n-1) + 3
a1 = 5
an = an-1 + 3
Problem 7
Write a recursive formula for 7, 21, 63, ….
Worked answer
a_1 = 7, a_n = 3a_(n-1)
a1 = 7
an = 3an-1
Problem 8
Write an explicit formula for 80, 40, 20, 10, ….
Worked answer
a_n = 80(1/2)^(n-1)
an = 80(12)n-1
Problem 9
An arithmetic sequence has a1 = -3 and d = 7. Find a20.
Worked answer
130
-3 + 19(7)
= -3 + 133 = 130
Problem 10
An arithmetic sequence has a3 = 11 and a7 = 31. Find d.
Worked answer
5
31 - 11 = 20
20 ÷ 4 = 5
Problem 11
The graph of a sequence is
- a set of separate points
- a continuous line
- a smooth curve
- a shaded region
Worked answer
a set of separate points
Only the counting numbers
So the graph is separate dots
Never join them
Problem 12
A geometric sequence has a1 = 5 and a4 = 135. Find r.
Worked answer
3
5r3 = 135
r3 = 27
r = 3
Problem 13
An arithmetic sequence has a1 = 100 and d = -6. Which term is the first negative one?
Worked answer
a_18
100 - 6n + 6 < 0
106 < 6n, so n > 17.67
The 18th term: 100 - 6(17) = -2
Problem 14
Write an explicit formula for the arithmetic sequence with a5 = 23 and a9 = 43.
Worked answer
a_n = 3 + 5(n - 1)
d = 43 - 234 = 5
a1 = 23 - 4(5) = 3
an = 3 + 5(n-1), or 5n - 2
Problem 15
How many terms are there in 7, 11, 15, …, 99?
Worked answer
24
(n-1)4 = 92
n - 1 = 23
n = 24
Problem 16
A geometric sequence has a2 = 12 and a5 = 324. Find a1.
Worked answer
4
12r3 = 324, so r3 = 27 and r = 3
a1 = 12 ÷ 3
= 4